On the Lukacs property for free random variables

Kamil Szpojankowski

Displaying similar documents to “On the Lukacs property for free random variables”

Multiplicative free square of the free Poisson measure and examples of free symmetrization

Melanie Hinz, Wojciech Młotkowski (2010)

Colloquium Mathematicae

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We compute the moments and free cumulants of the measure $ρ_{t} : = π_{t} ⊠ π_{t}$ , where $π_{t}$ denotes the free Poisson law with parameter t > 0. We also compute free cumulants of the symmetrization of $ρ_{t}$ . Finally, we introduce the free symmetrization of a probability measure on ℝ and provide some examples.

Free powers of the free Poisson measure

Melanie Hinz, Wojciech Młotkowski (2011)

Colloquium Mathematicae

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We compute moments of the measures ${(ϖ^{⊠ p})}^{⊞ t}$ , where ϖ denotes the free Poisson law, and ⊞ and ⊠ are the additive and multiplicative free convolutions. These moments are expressed in terms of the Fuss-Narayana numbers.

On graphs G for which both g and G̅ are claw-free

Shinya Fujita (2005)

Discussiones Mathematicae Graph Theory

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Let G be a graph with |V(G)| ≥ 10. We prove that if both G and G̅ are claw-free, then minΔ(G), Δ(G̅) ≤ 2. As a generalization of this result in the case where |V(G)| is sufficiently large, we also prove that if both G and G̅ are $K_{1, t}$ -free, then minΔ(G),Δ(G̅) ≤ r(t- 1,t)-1 where r(t-1,t) is the Ramsey number.

The square-free kernel of $x^{2^{n}} - a^{2^{n}}$

Paulo Ribenboim (2002)

Acta Arithmetica

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A free stochastic partial differential equation

Yoann Dabrowski (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We get stationary solutions of a free stochastic partial differential equation. As an application, we prove equality of non-microstate and microstate free entropy dimensions under a Lipschitz like condition on conjugate variables, assuming also the von Neumann algebra $R^{ω}$ embeddable. This includes an $N$ -tuple of $q$ -Gaussian random variables e.g. for $| q | N \leq 0.13$ .

$ℵ_{k}$ -free separable groups with prescribed endomorphism ring

Daniel Herden, Héctor Gabriel Salazar Pedroza (2015)

Fundamenta Mathematicae

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We will consider unital rings A with free additive group, and want to construct (in ZFC) for each natural number k a family of $ℵ_{k}$ -free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is $ℵ_{k}$ -free if every subset of size $< ℵ_{k}$ is contained in a free submodule (we will refine this in Definition 3.2); and it is separable as an abelian group if any finite subset of G is contained in a free direct summand of G. Despite the fact that such a...

On fixed point free involutions of $R^{1} \times S^{2}$

Gerhard X. Ritter (1976)

Colloquium Mathematicae

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Large free subgroups of automorphism groups of ultrahomogeneous spaces

Szymon Głąb, Filip Strobin (2015)

Colloquium Mathematicae

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We consider the following notion of largeness for subgroups of $S_{\infty}$ . A group G is large if it contains a free subgroup on generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of $S_{\infty}$ can be extended to a large free subgroup of $S_{\infty}$ , and, under Martin’s Axiom, any free subgroup of $S_{\infty}$ of cardinality less than can also be extended to a large free subgroup of $S_{\infty}$ . Finally, if Gₙ are countable...

Computation of some examples of Brown's spectral measure in free probability

Philippe Biane, Franz Lehner (2001)

Colloquium Mathematicae

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We use free probability techniques to compute spectra and Brown measures of some non-hermitian operators in finite von Neumann algebras. Examples include $u ₙ + u_{\infty}$ where uₙ and $u_{\infty}$ are the generators of ℤₙ and ℤ respectively, in the free product ℤₙ*ℤ, or elliptic elements of the form $S_{α} + i S_{β}$ where $S_{α}$ and $S_{β}$ are free semicircular elements of variance α and β.

On some free semigroups, generated by matrices

Piotr Słanina (2015)

Czechoslovak Mathematical Journal

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Let $A = [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}], B_{λ} = [\begin{matrix} 1 & 0 \\ λ & 1 \end{matrix}] .$ We call a complex number $λ$ “semigroup free“ if the semigroup generated by $A$ and $B_{λ}$ is free and “free” if the group generated by $A$ and $B_{λ}$ is free. First families of semigroup free $λ$ ’s were described by J. L. Brenner, A. Charnow (1978). In this paper we enlarge the set of known semigroup free $λ$ ’s. To do it, we use a new version of “Ping-Pong Lemma” for semigroups embeddable in groups. At the end we present most of the known results related to semigroup free and free numbers in a common...

Regularity of Lipschitz free boundaries for the thin one-phase problem

Daniela De Silva, Ovidiu Savin (2015)

Journal of the European Mathematical Society

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We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $E (u, Ω) = \int_{Ω} {| \nabla u |}^{2} d X + ℋ^{n} ({u > 0} \cap {x_{n + 1} = 0}), Ω \subset ℝ^{n + 1},$ among all functions $u \geq 0$ which are fixed on $\partial Ω$ .

Spectral distribution of the free Jacobi process associated with one projection

Nizar Demni, Taoufik Hmidi (2014)

Colloquium Mathematicae

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Given an orthogonal projection P and a free unitary Brownian motion $Y = {(Y ₜ)}_{t \geq 0}$ in a W*-non commutative probability space such that Y and P are *-free in Voiculescu’s sense, we study the spectral distribution νₜ of Jₜ = PYₜPYₜ*P in the compressed space. To this end, we focus on the spectral distribution μₜ of the unitary operator SYₜSYₜ*, S = 2P - 1, whose moments are related to those of Jₜ via a binomial-type expansion already obtained by Demni et al. [Indiana Univ. Math. J. 61 (2012)]. In this...

A note on certain partial sum operators

Marek Bożejko, Gero Fendler (2006)

Banach Center Publications

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We show that for the t-deformed semicircle measure, where 1/2 < t ≤ 1, the expansions of $L_{p}$ functions with respect to the associated orthonormal polynomials converge in norm when 3/2 < p < 3 and do not converge when 1 ≤ p < 3/2 or 3 < p. From this we conclude that natural expansions in the non-commutative $L_{p}$ spaces of free group factors and of free commutation relations do not converge for 1 ≤ p < 3/2 or 3 < p.

On the global regularity of subcritical Euler–Poisson equations with pressure

Eitan Tadmor, Dongming Wei (2008)

Journal of the European Mathematical Society

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We prove that the one-dimensional Euler–Poisson system driven by the Poisson forcing together with the usual $γ$ -law pressure, $γ \geq 1$ , admits global solutions for a large class of initial data. Thus, the Poisson forcing regularizes the generic finite-time breakdown in the $2 \times 2$ $p$ -system. Global regularity is shown to depend on whether or not the initial configuration of the Riemann invariants and density crosses an intrinsic critical threshold.

Formally self-referential propositions for cut free classical analysis and related systems

G. Kreisel, G. Takeuti

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CONTENTSIntroduction............................................................................................................................................................................................................ 5 I. Results on self-referential propositions............................................................................................................................. 11 1. Definitions of some principal metamathematical notions......................................................................

Maps into the torus and minimal coincidence sets for homotopies

D. L. Goncalves, M. R. Kelly (2002)

Fundamenta Mathematicae

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Let X,Y be manifolds of the same dimension. Given continuous mappings $f_{i}, g_{i} : X \to Y$ , i = 0,1, we consider the 1-parameter coincidence problem of finding homotopies $f_{t}, g_{t}$ , 0 ≤ t ≤ 1, such that the number of coincidence points for the pair $f_{t}, g_{t}$ is independent of t. When Y is the torus and f₀,g₀ are coincidence free we produce coincidence free pairs f₁,g₁ such that no homotopy joining them is coincidence free at each level. When X is also the torus we characterize the solution of the problem in terms of the...

Maximal free sequences in a Boolean algebra

J. D. Monk (2011)

Commentationes Mathematicae Universitatis Carolinae

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We study free sequences and related notions on Boolean algebras. A free sequence on a BA $A$ is a sequence $〈 a_{ξ} : ξ < α 〉$ of elements of $A$ , with $α$ an ordinal, such that for all $F, G \in {[α]}^{< ω}$ with $F < G$ we have $\prod_{ξ \in F} a_{ξ} \cdot \prod_{ξ \in G} - a_{ξ} \neq 0$ . A free sequence of length $α$ exists iff the Stone space $Ult (A)$ has a free sequence of length $α$ in the topological sense. A free sequence is maximal iff it cannot be extended at the end to a longer free sequence. The main notions studied here are the spectrum function $𝔣_{sp} (A) = {| α | : A has an infinite maximal free sequence of length α}$ and the associated min-max function $𝔣 (A) = min (𝔣_{sp} (A)) .$ Among...

The Herz-Schur multiplier norm of sets satisfying the Leinert condition

Éric Ricard, Ana-Maria Stan (2011)

Colloquium Mathematicae

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It is well known that in a free group , one has $| | χ_{E} {| |}_{M_{c b} A ()} \leq 2$ , where E is the set of all the generators. We show that the (completely) bounded multiplier norm of any set satisfying the Leinert condition depends only on its cardinality. Consequently, based on a result of Wysoczański, we obtain a formula for $| | χ_{E} {| |}_{M_{c b} A ()}$ .

Elementary embeddings in torsion-free hyperbolic groups

Chloé Perin (2011)

Annales scientifiques de l'École Normale Supérieure

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We describe first-order logic elementary embeddings in a torsion-free hyperbolic group in terms of Sela’s hyperbolic towers. Thus, if $H$ embeds elementarily in a torsion free hyperbolic group $Γ$ , we show that the group $Γ$ can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of $H$ with some free group and groups of closed surfaces. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. We also...

Mean field limit for the one dimensional Vlasov-Poisson equation

Maxime Hauray (2012-2013)

Séminaire Laurent Schwartz — EDP et applications

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We consider systems of $N$ particles in dimension one, driven by pair Coulombian or gravitational interactions. When the number of particles goes to infinity in the so called mean field scaling, we formally expect convergence towards the Vlasov-Poisson equation. Actually a rigorous proof of that convergence was given by Trocheris in [Tro86]. Here we shall give a simpler proof of this result, and explain why it implies the so-called “Propagation of molecular chaos”. More precisely, both...